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July  2020, 16(4): 1663-1683. doi: 10.3934/jimo.2019023

Minimizing almost smooth control variation in nonlinear optimal control problems

Ying Zhang 1, , Changjun Yu 2,, , Yingtao Xu 3, and  Yanqin Bai 2,

1. 

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, China
2. 

Department of Mathematics, Shanghai University, Baoshan 200444, Shanghai, China
3. 

Xingzhi College, Zhejiang Normal University, Jinhua 321004, Zhejiang, China

* Corresponding author: Changjun Yu

Received  January 2018 Revised  March 2018 Published  March 2019

Fund Project: This paper is supported by NSFC grant 11871039, 11771275, the Scientific Research Project of Zhejiang Provincial Department of science and technology in China (Grant No. LGN19C040001), and the Scientific Research Project of Zhejiang Provincial Department of Education in China (Grant No. Y201329106)

In this paper, we consider an optimal control problem in which the control is almost smooth and the state and control are subject to terminal state constraints and continuous state and control inequality constraints. By introducing an extra set of differential equations for this almost smooth control, we transform this constrained optimal control problem into an equivalent problem involving both control function and system parameter vector as decision variables. Then, by the control parametrization technique and a time scaling transformation, the equivalent problem is approximated by a sequence of constrained optimal parameter selection problems, each of which is a finite dimensional optimization problem. For each of these constrained optimal parameter selection problems, a novel exact penalty function method is constructed by appending penalized constraint violations to the cost function. This gives rise to a sequence of unconstrained optimal parameter selection problems; and each of which can be solved by existing optimization algorithms or software packages. Finally, a practical container crane operation problem is solved, showing the effectiveness and applicability of the proposed approach.

Keywords: Optimal control, control variation, almost smooth control.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Ying Zhang, Changjun Yu, Yingtao Xu, Yanqin Bai. Minimizing almost smooth control variation in nonlinear optimal control problems. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1663-1683. doi: 10.3934/jimo.2019023
References:
[1]

N. U. Ahmed, Dynamic Systems and Control with Applications, Singapore: World Scientific, 2006. doi: 10.1142/6262.  Google Scholar

[2]

N. Banihashemi and C. Y. Kaya, Inexact restoration and adaptive mesh refinement for optimal control, Journal of Industrial and Management Optimization, 10 (2014), 521-542.  doi: 10.3934/jimo.2014.10.521.  Google Scholar

[3]

D. Chang and Z. Wu, Stochastic maximum principle for non-zero sum differential games of FBSDEs with impulse controls and its application to finance, Journal of Industrial and Management Optimization, 11 (2015), 27-40.  doi: 10.3934/jimo.2015.11.27.  Google Scholar

[4]

M. Gerdts and M. Kunkel, A nonsmooth Newton's method for discretized optimal control problems with state and control constraints, Journal of Industrial and Management Optimization, 4 (2008), 247-270.  doi: 10.3934/jimo.2008.4.247.  Google Scholar

[5]

L. Göllmann and H. Maurer, Theory and applications of optimal control problems with multiple time-delays, Journal of Industrial and Management Optimization, 10 (2014), 413-441.  doi: 10.3934/jimo.2014.10.413.  Google Scholar

[6]

Y. Han and Y. Gao, Determining the viability for hybrid control systems on a region with piecewise smooth boundary, Numerical Algebra, Control and Optimization, 5 (2015), 1-9.  doi: 10.3934/naco.2015.5.1.  Google Scholar

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L. S. Jennings, M. E. Fisher, K. L. Teo, et al., MISER 3 Optimal Control Software: Theory and User Manual, version 3, University of Western Australia, 2004. Google Scholar

[8]

L. Jennings, C. Yu, B. Li, V. Rehbock, R. Loxton and F. Yang, Visual miser: an efficient user-friendly visual program for solving optimal control problems, Journal of Industrial and Management Optimization, 12 (2015), 781-810.  doi: 10.3934/jimo.2016.12.781.  Google Scholar

[9]

J. Kaartinen, J. Hätönen, H. Hyötyniemi and J. Miettunen, Machine-vision-based control of zinc flotation: A case study, Control Engineering Practice, 14 (2006), 1455-1466.  doi: 10.1016/j.conengprac.2005.12.004.  Google Scholar

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C. T. Lawrence and A. L. Tits, A computationally efficient feasible sequential quadratic programming algorithm, SIAM Journal on Optimization, 11 (2006), 1092-1118.  doi: 10.1137/S1052623498344562.  Google Scholar

[11]

H. W. J. Lee, K. L. Teo, V. Rehbock and L. S. Jennings, Control parametrization enhancing technique for time optimal control problems, Dynamic Systems and Applications, 6 (1997), 243-261.   Google Scholar

[12]

B. Li, C. Xu, K. L. Teo and J. Chu, Time optimal Zermelo's navigation problem with moving and fixed obstacles, Applied Mathematics and Computation, 224 (2013), 866-875.  doi: 10.1016/j.amc.2013.08.092.  Google Scholar

[13]

Q. Lin, R. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A surney, Journal of Industrial and Management Optimization, 10 (2014), 275-309.  doi: 10.3934/jimo.2014.10.275.  Google Scholar

[14]

R. Loxton, Q. Lin, V. Rehbock and K. L. Teo, Control parameterization for optimal control problems with continuous inequality constraints: New convergence results, Numerical Algebra, Control and Optimization, 2 (2012), 571-599.  doi: 10.3934/naco.2012.2.571.  Google Scholar

[15]

K. S. Peterson and A. G. Stefanopoulou, Extremum seeking control for soft landing of an electromechanical valve actuator, Automatica, 40 (2004), 1063-1069.  doi: 10.1016/j.automatica.2004.01.027.  Google Scholar

[16]

V. Rehbock, Tracking Control and Optimal Control, PhD thesis, University of Western Australia, Perth, 1994. Google Scholar

[17]

Y. Sakawa and Y. Shindo, Optimal control of container cranes, Automatica, 18 (1982), 257-266.   Google Scholar

[18]

K. Schittkowski, NLPQL: A fortran subroutine solving constrained nonlinear programming problems, Ann. Oper. Res., 5 (1986), 485-500.  doi: 10.1007/BF02739235.  Google Scholar

[19]

K. L. Teo and L. S. Jennings, Nonlinear optimal control problems with continuous state inequality constraints, Journal of Optimization Theory and Application, 63 (1989), 1-22.  doi: 10.1007/BF00940727.  Google Scholar

[20]

K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Long-man Scientific and Technical, Essex, 1991.  Google Scholar

[21]

W. X. Wang, Y. L. Shang, L. S. Zhang, et. al., Global minimization of non-smooth unconstrained problems with filled function, Optimization Letters, 7 (2013), 435-446. doi: 10.1007/s11590-011-0427-7.  Google Scholar

[22]

W. Xu, Z. G. Feng, J. W. Peng and K. F. C. Yiu, Optimal switching for linear quadratic problem of switched systems in discrete time, Automatica, 78 (2017), 185-193.  doi: 10.1016/j.automatica.2016.12.002.  Google Scholar

[23]

K. F. C. Yiu, Y. Liu and K. L. Teo, A hybrid descent method for global optimization, Journal of Global Optimization, 28 (2004), 229-238.  doi: 10.1023/B:JOGO.0000015313.93974.b0.  Google Scholar

[24]

C. Yu, K. L. Teo, L. S. Zhang and Y. Q. Bai, A new exact penalty function method for continuous inequality constrained optimization problems, Journal of Industrial and Management Optimization, 6 (2010), 895-910.  doi: 10.3934/jimo.2010.6.895.  Google Scholar

[25]

C. Yu, K. L. Teo, L. S. Zhang and Y. Q. Bai, On a refinement of the convergence analysis for the new exact penalty function method for continuous inequality constrained optimization problem, Journal of Industrial Management and Optimization, 8 (2012), 485-491.  doi: 10.3934/jimo.2012.8.485.  Google Scholar

show all references

References:
[1]

N. U. Ahmed, Dynamic Systems and Control with Applications, Singapore: World Scientific, 2006. doi: 10.1142/6262.  Google Scholar

[2]

N. Banihashemi and C. Y. Kaya, Inexact restoration and adaptive mesh refinement for optimal control, Journal of Industrial and Management Optimization, 10 (2014), 521-542.  doi: 10.3934/jimo.2014.10.521.  Google Scholar

[3]

D. Chang and Z. Wu, Stochastic maximum principle for non-zero sum differential games of FBSDEs with impulse controls and its application to finance, Journal of Industrial and Management Optimization, 11 (2015), 27-40.  doi: 10.3934/jimo.2015.11.27.  Google Scholar

[4]

M. Gerdts and M. Kunkel, A nonsmooth Newton's method for discretized optimal control problems with state and control constraints, Journal of Industrial and Management Optimization, 4 (2008), 247-270.  doi: 10.3934/jimo.2008.4.247.  Google Scholar

[5]

L. Göllmann and H. Maurer, Theory and applications of optimal control problems with multiple time-delays, Journal of Industrial and Management Optimization, 10 (2014), 413-441.  doi: 10.3934/jimo.2014.10.413.  Google Scholar

[6]

Y. Han and Y. Gao, Determining the viability for hybrid control systems on a region with piecewise smooth boundary, Numerical Algebra, Control and Optimization, 5 (2015), 1-9.  doi: 10.3934/naco.2015.5.1.  Google Scholar

[7]

L. S. Jennings, M. E. Fisher, K. L. Teo, et al., MISER 3 Optimal Control Software: Theory and User Manual, version 3, University of Western Australia, 2004. Google Scholar

[8]

L. Jennings, C. Yu, B. Li, V. Rehbock, R. Loxton and F. Yang, Visual miser: an efficient user-friendly visual program for solving optimal control problems, Journal of Industrial and Management Optimization, 12 (2015), 781-810.  doi: 10.3934/jimo.2016.12.781.  Google Scholar

[9]

J. Kaartinen, J. Hätönen, H. Hyötyniemi and J. Miettunen, Machine-vision-based control of zinc flotation: A case study, Control Engineering Practice, 14 (2006), 1455-1466.  doi: 10.1016/j.conengprac.2005.12.004.  Google Scholar

[10]

C. T. Lawrence and A. L. Tits, A computationally efficient feasible sequential quadratic programming algorithm, SIAM Journal on Optimization, 11 (2006), 1092-1118.  doi: 10.1137/S1052623498344562.  Google Scholar

[11]

H. W. J. Lee, K. L. Teo, V. Rehbock and L. S. Jennings, Control parametrization enhancing technique for time optimal control problems, Dynamic Systems and Applications, 6 (1997), 243-261.   Google Scholar

[12]

B. Li, C. Xu, K. L. Teo and J. Chu, Time optimal Zermelo's navigation problem with moving and fixed obstacles, Applied Mathematics and Computation, 224 (2013), 866-875.  doi: 10.1016/j.amc.2013.08.092.  Google Scholar

[13]

Q. Lin, R. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A surney, Journal of Industrial and Management Optimization, 10 (2014), 275-309.  doi: 10.3934/jimo.2014.10.275.  Google Scholar

[14]

R. Loxton, Q. Lin, V. Rehbock and K. L. Teo, Control parameterization for optimal control problems with continuous inequality constraints: New convergence results, Numerical Algebra, Control and Optimization, 2 (2012), 571-599.  doi: 10.3934/naco.2012.2.571.  Google Scholar

[15]

K. S. Peterson and A. G. Stefanopoulou, Extremum seeking control for soft landing of an electromechanical valve actuator, Automatica, 40 (2004), 1063-1069.  doi: 10.1016/j.automatica.2004.01.027.  Google Scholar

[16]

V. Rehbock, Tracking Control and Optimal Control, PhD thesis, University of Western Australia, Perth, 1994. Google Scholar

[17]

Y. Sakawa and Y. Shindo, Optimal control of container cranes, Automatica, 18 (1982), 257-266.   Google Scholar

[18]

K. Schittkowski, NLPQL: A fortran subroutine solving constrained nonlinear programming problems, Ann. Oper. Res., 5 (1986), 485-500.  doi: 10.1007/BF02739235.  Google Scholar

[19]

K. L. Teo and L. S. Jennings, Nonlinear optimal control problems with continuous state inequality constraints, Journal of Optimization Theory and Application, 63 (1989), 1-22.  doi: 10.1007/BF00940727.  Google Scholar

[20]

K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Long-man Scientific and Technical, Essex, 1991.  Google Scholar

[21]

W. X. Wang, Y. L. Shang, L. S. Zhang, et. al., Global minimization of non-smooth unconstrained problems with filled function, Optimization Letters, 7 (2013), 435-446. doi: 10.1007/s11590-011-0427-7.  Google Scholar

[22]

W. Xu, Z. G. Feng, J. W. Peng and K. F. C. Yiu, Optimal switching for linear quadratic problem of switched systems in discrete time, Automatica, 78 (2017), 185-193.  doi: 10.1016/j.automatica.2016.12.002.  Google Scholar

[23]

K. F. C. Yiu, Y. Liu and K. L. Teo, A hybrid descent method for global optimization, Journal of Global Optimization, 28 (2004), 229-238.  doi: 10.1023/B:JOGO.0000015313.93974.b0.  Google Scholar

[24]

C. Yu, K. L. Teo, L. S. Zhang and Y. Q. Bai, A new exact penalty function method for continuous inequality constrained optimization problems, Journal of Industrial and Management Optimization, 6 (2010), 895-910.  doi: 10.3934/jimo.2010.6.895.  Google Scholar

[25]

C. Yu, K. L. Teo, L. S. Zhang and Y. Q. Bai, On a refinement of the convergence analysis for the new exact penalty function method for continuous inequality constrained optimization problem, Journal of Industrial Management and Optimization, 8 (2012), 485-491.  doi: 10.3934/jimo.2012.8.485.  Google Scholar

Figure 1.  Optimal control $ u_{1}(t) $.
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Figure 2.  Optimal control function $ u_{2}(t) $.
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Figure 3.  Optimal state trajectory $ x_{1}(t) $.
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Figure 4.  Optimal state trajectory $ x_{2}(t) $.
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Figure 5.  Optimal control $ x_{3}(t) $.
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Figure 6.  Optimal state trajectory $ x_{4}(t) $.
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Figure 7.  Optimal state trajectory $ x_{5}(t) $.
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Figure 8.  Optimal control $ x_{6}(t) $.
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